Exponentially fitted symplectic methods for the numerical integration of the Schrödinger equation

The computation of the energy eigenvalues of the one-dimensional time-independent Schrödinger equation is considered. Exponentially fitted and trigonometrically fitted symplectic integrators are obtained, by modification of the first and second order Yoshida symplectic methods. Numerical results are obtained for the one-dimensional harmonic oscillator and Morse potential.AMS subject classification: 65L15Funding by research project 71239 of Prefecture of Western Macedonia and the E.U. is gratefully acknowledged.     

New fifth‐order two‐derivative Runge‐Kutta methods with constant and frequency‐dependent coefficients

Two‐derivative Runge‐Kutta methods are Runge‐Kutta methods for problems of the form y′ = f(y) that include the second derivative y′′ = g(y) = f ′(y)f(y) and were developed in the work of Chan and Tsai. In this work, we consider explicit methods and construct a family of fifth‐order methods with three stages of the general case that use several evaluations of f and g per step. For problems with oscillatory solution and in the case that a good estimate of the dominant frequency is known, methods with frequency‐dependent coefficients are used; there are several procedures for constructing such methods. We give the general framework for the construction of methods with variable coefficients following the approach of Simos. We modify the above family to derive methods with frequency‐dependent coefficients following this approach as well as the approach given by Vanden Berghe. We provide numerical results to demonstrate the efficiency of the new methods using three test problems.     

Multicompartmental Microcapsules with Orthogonal Programmable Two‐Way Sequencing of Hydrophobic and Hydrophilic Cargo Release

Multicompartmental responsive microstructures with the capability for the pre‐programmed sequential release of multiple target molecules of opposite solubility (hydrophobic and hydrophilic) in a controlled manner have been fabricated. Star block copolymers with dual‐responsive blocks (temperature for poly(N‐isopropylacrylamide) chains and pH for poly(acrylic acid) and poly(2‐vinylpyridine) arms) and unimolecular micellar structures serve as nanocarriers for hydrophobic molecules in the microcapsule shell. The interior of the microcapsule can be loaded with water‐soluble hydrophilic macromolecules. For these dual‐loaded microcapsules, a programmable and sequential release of hydrophobic and hydrophilic molecules from the shell and core, respectively, can be triggered independently by temperature and pH variations. These stimuli affect the hydrophobicity and chain conformation of the star block copolymers to initiate out‐of‐shell release (elevated temperature), or change the overall star conformation and interlayer interactions to trigger increased permeability of the shell and out‐of‐core release (pH). Reversing stimulus order completely alters the release process.     

Computation of the eigenvalues of the Schrödinger equation by symplectic and trigonometrically fitted symplectic partitioned Runge-Kutta methods

In this Letter we present an explicit symplectic method for the numerical solution of the Schrödinger equation. We also develop a modified symplectic integrator with the trigonometrically fitted property based on this method. Our new methods are tested on the computation of the eigenvalues of the one-dimensional harmonic oscillator, the doubly anharmonic oscillator and the Morse potential.     

pH responsive self assemblies from an A(n)-core-(B-b-C)n heteroarm star block terpolymer bearing oppositely charged segments

We report on the association capability of a novel multisegmented, multiarm star terpolymer to form a diversity of pH-responsive amphoteric micellar nanostructured self-assemblies.     

Design of “smart” segmented polymers by incorporating random copolymers as building blocks

This article is dealing with the design of novel segmented polymers comprising homopolymer and random copolymer building blocks designated as block-random. This type of polymeric materials can be prepared through macromolecular engineering by using controlled polymerization methods. By replacing a homopolymer block with a random one, in block copolymer topologies, further tuning of the copolymer properties can be achieved. The present article highlights the recent developments on block-random segmented macromolecules, bearing building blocks of tunable properties (e.g. thermo-sensitivity (LCST), hydrophobicity) and exhibiting responsive behavior in aqueous environments. Furthermore, preliminary novel results regarding pH-sensitive segmented macromolecules of various topologies, bearing random polyampholyte blocks among others, are also demonstrated and discussed.     

Two-derivative Runge-Kutta methods with optimal phase properties

In this work, we consider two-derivative Runge-Kutta methods for the numerical integration of first-order differential equations with oscillatory solution. We construct methods with constant coefficients and special properties as minimum phase-lag and amplification errors with three and four stages. All methods constructed have fifth algebraic order. We also present methods with variable coefficients with zero phase-lag and amplification errors. In order to examine the efficiency of the new methods, we use four well-known oscillatory test problems.